Problem 37 - Precalculus Essentials

Question

In Exercises \(35-38,\) use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. $$\sin ^{2} x \cos ^{2} x$$

Step-by-Step Solution

Answer

The expression \(\sin ^{2} x \cos ^{2} x\) simplifies to \(1/4 \sin^{2}(2x)\) using the power-reducing formulas.

Step 1: Identify the power reducing formulas for sine and cosine

The power reducing formulas that will be used are: \(\sin^{2}(x) = 1/2(1 - \cos(2x))\) and \(\cos^{2}(x) = 1/2(1 + \cos(2x))\). These formulas help in reducing the power of the trigonometric functions to 1.

Step 2: Substitute in the power reducing formulas

Using the power reducing formulas, the expression \(\sin ^{2} x \cos ^{2} x\) is substituted to obtain (1/2(1 - \cos(2x))) * (1/2(1 + \cos(2x))).

Step 3: Simplify the result

On multiplication, the resulting expression is \(1/4(1-\cos^{2}(2x))\). Now apply the Pythagorean identity \(\sin^{2}(a) + \cos^{2}(a) = 1\), where \(a = 2x\). The expression can be rewritten as \(1/4(1 -(1- \sin^{2}(2x)))\). Further simplification gives \(1/4 \sin^{2}(2x)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that relate different trigonometric functions to each other. These identities are incredibly useful in simplifying complex expressions and solving trigonometric equations. The power-reducing formulas mentioned in the original exercise are part of these identities.
Power-reducing formulas are specifically used to express trigonometric functions raised to higher powers, such as \( \sin^{2}(x) \) and \( \cos^{2}(x) \), in terms of first powers. When working with these identities, the key is replacing higher powers of sines and cosines with expressions that involve simpler functions.
This is achieved through the following:

• The sine power-reducing formula: \( \sin^{2}(x) = \frac{1}{2}(1 - \cos(2x)) \)
• The cosine power-reducing formula: \( \cos^{2}(x) = \frac{1}{2}(1 + \cos(2x)) \)

Understanding these formulas and the purpose of using various trigonometric identities will enable you to manipulate and simplify expressions efficiently and accurately.

Expression Simplification

Expression simplification is the process of making mathematical expressions easier to understand or solve. In the context of trigonometry, this often involves using identities to rewrite expressions so they are in their simplest form, reducing the likelihood for error in further calculations.
For the given example, the task is to simplify \( \sin^{2} x \cos^{2} x \) using the power-reducing formulas. Initially, both powers of sine and cosine are replaced using these identities. The expression becomes a product of two terms: \[ \left( \frac{1}{2} (1 - \cos(2x)) \right) \times \left( \frac{1}{2} (1 + \cos(2x)) \right) \]
After substitution, multiplication of these terms simplifies the expression. Simplifying requires distribution and combining like terms or using formulas that can make the expressions more manageable. This basic idea is crucial not only in trigonometry but in all areas of mathematics where simplification lends clarity to complex problems.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental relationships in trigonometry. It relates the squares of sine and cosine to 1. The identity is expressed as:

• \( \sin^{2}(a) + \cos^{2}(a) = 1 \)

This identity arises from the Pythagorean theorem in a unit circle context and is invaluable when dealing with expressions involving squared trigonometric functions.
In the exercise, after simplifying the product of power-reduced expressions, it resulted in \( 1/4(1 - \cos^{2}(2x)) \). To simplify further, the Pythagorean identity is used: \( \cos^{2}(2x) = 1 - \sin^{2}(2x) \).
This substitution transforms the expression into \( 1/4 \sin^{2}(2x) \), achieving the goal of reducing all trigonometric powers to one. Understanding and using the Pythagorean identity helps bridge the gap between complex expressions and their simpler forms, making trigonometry more approachable.